ISSN 2307-7743 http://scienceasia.asia
SOME FIXED POINT THEOREMS FOR GENERALIZED
CONTRACTION INVOLVING RATIONAL EXPRESSIONS IN COMPLEX VALUED B-METRIC SPACES
ANIL KUMAR DUBEY1,∗, MANJULA TRIPATHI2, M.D. PANDEY1
1Department of Mathematics, Bhilai Institute of Technology, Bhilai House, Durg,
Chhattisgarh 491001, India
2Department of Mathematics, U.P.U Government Polytechnic, Durg, Chhattisgarh 491001,
India
∗Correspondence: [email protected]
Abstract. The purpose of this paper is to prove the existence and uniqueness of a common
fixed point for a pair of mappings satisfying generalized contraction under rational expres-sions having point-dependent control functions as coefficients in complex valued b-metric
spaces. Our results generalize and extend the results of Azam et al., Dass et al., Dubey et al., Mukheimer, Nashine et al., Rao et al. and Sitthikul et al.
1. Introduction and Preliminaries
Complex valued metric spaces were introduced in 2011 by Azam et al. [1] and established the existence of fixed point theorems for maps satisfying the contraction condition involving rational expressions. Several fixed point results in such spaces were obtained, for example, in [2,8,9,11,12,13,14]. The concept of complex valued b-metric space as a generalization of complex valued metric space .[1] was initiated by Rao et al. [10]. Subsequently, many authors proved fixed and common fixed point results in complex valued b-metric spaces [4,5,6,7].
The aim of this paper is to give the development of fixed point theorems on complex valued b-metric space. We prove fixed and common fixed point results for generalized contraction involving rational expressions in complex valued b-metric space which generalize all the existing results. In particular, complex valued b-metric type version of very well known results of Nashine et al. [9], Sitthikul et al.[12] and others are obtained.
Keywords: Common fixed point, Complex valued b-metric spaces, Complete complex valued b-metric spaces, Cauchy sequence, Fixed point.
c
In what follows, we recall some notations and definitions due to Rao et al. [10], that will be needed in our subsequent discussion.
LetCbe the set of complex numbers andz1, z2 ∈C.Define a partial order-onCas follows:
z1 -z2 if and only ifRe(z1)≤Re(z2), Im(z1)≤Im(z2).
Consequently, one can infer that z1 -z2 if one of the following conditions is satisfied:
(i) Re(z1) =Re(z2), Im(z1)< Im(z2), (ii) Re(z1)< Re(z2), Im(z1) =Im(z2), (iii) Re(z1)< Re(z2), Im(z1)< Im(z2), (iv) Re(z1) =Re(z2), Im(z1) = Im(z2).
In particular, we writez1 z2 ifz1 6=z2 and one of (i),(ii) and (iii) is satisfied, also we write
z1 ≺z2 if only (iii) is satisfied. Notice that
(a) If 0-z1 z2 then |z1|<|z2|. (b) Ifz1 -z2 and z2 ≺z3 then z1 ≺z3.
(c) Ifa, b∈Rand a≤b then az -bz for all z ∈C.
The following definition is recently introduced by Rao et al. [10].
Definition 1.1[10] Let X be a nonempty set and let s ≥ 1 be a given real number. A function d :X×X →C is called a complex valued b-metric on X if for all x, y, z ∈ X the following conditions are satisfied:
(i) 0-d(x, y) and d(x, y) = 0 if and only if x=y; (ii) d(x, y) =d(y, x);
(iii) d(x, y)-s[d(x, z) +d(z, y)].
The pair (X, d) is called a complex valued b-metric space.
Example 1.2[10] Let X = [0,1]. Define the mapping d : X ×X → C by d(x, y) =
|x−y|2 +i|x−y|2, for all x, y ∈ X. Then (X, d) is a complex valued b-metric space with
Definition 1.3[10] Let (X, d) be a complex valued b-metric space.
(i) A point x ∈ X is called interior point of a set A ⊆ X whenever there exists 0 ≺ r ∈ C
such that B(x, r) ={y ∈X:d(x, y)≺r} ⊆A.
(ii) A point x∈X is called a limit point of a setA whenever for every 0≺r∈C, B(x, r)∩
(A− {x})6=φ.
(iii) A subset A⊆X is called open whenever each element of A is an interior point of a set
A.
(iv) A subset A⊆X is called closed whenever each element of A belongs to A.
(v) A sub-basis for a Hausdorff topologyτ onX is a familyF ={B(x, r) :x∈X and0≺r}.
Definition 1.4[10] Let (X, d) be a complex valued b-metric space, {xn} be a sequence in
X and x∈X.
(i) If for every c ∈ C, with 0 ≺ r there is N ∈ N such that for all n > N, d(xn, x) ≺ c,
then {xn} is said to be convergent, {xn}converges to xand x is the limit point of {xn}. We
denote this by limn→∞xn=x or{xn} →x as n→ ∞.
(ii) If for every c∈ C, with 0≺r there is N ∈N such that for all n > N, d(xn, xn+m)≺ c,
where m∈N, then {xn}is said to be Cauchy sequence.
(iii) If every Cauchy sequence inXis convergent, then (X, d) is said to be a complete complex valued b-metric space.
Lemma 1.5[10] Let (X, d) be a complex valued b-metric space and let {xn}be a sequence
inX. Then {xn}converges to x if and only if |d(xn, x)| →0 as n → ∞.
Lemma 1.6[10]Let (X, d) be a complex valued b-metric space and let {xn}be a sequence
in X. Then {xn} is a Cauchy sequence if and only if |d(xn, xn+m)| → 0 as n → ∞, where
2. Main Result
We start to this section with the following observation, that will be used in our subsequent result.
Proposition 2.1. Let (X, d) be a complex valued b-metric space and S, T : X → X. Let
x0 ∈X and defined the sequence{xn} by
x2n+1 =Sx2n
x2n+2 =T x2n+1, for all n = 0,1,2,− − −. (2.1)
Assume that there exists a mapping ∝: X ×X → [0,1) such that ∝ (T Sx, y) ≤∝ (x, y) and ∝(x, ST y)≤∝(x, y) for all x, y ∈X. Then ∝(x2n, y)≤∝(x0, y) and ∝(x, x2n+1)≤∝ (x, x1) for all x, y ∈X and n= 0,1,2,− − −.
Proof. Letx, y ∈X and n = 0,1,2,− − −. Then we have
∝(x2n, y) =∝(T x2n−1, y) =∝(T Sx2n−2, y)
≤∝(x2n−2, y)
=∝(T x2n−3, y) =∝(T Sx2n−4, y)
≤∝(x2n−4, y)≤ − − −− ≤∝(x0, y).
Similarly, we have
∝(x, x2n+1) =∝(x, Sx2n) =∝(x, ST x2n−1)
≤∝(x, x2n−1)
=∝(x, Sx2n−2) =∝(x, ST x2n−3)
Lemma 2.2[12] Let {xn} be a sequence in X and h ∈[0,1). If an =|d(xn, xn+1)| satisfies
an ≤han−1,for all n∈N, then {xn} is a Cauchy sequence.
Now, we are ready to prove the following fixed point theorems for generalized contractions.
Theorem 2.3. Let (X, d) be a complete complex valued b-metric space with the coefficient
s ≥ 1 and let S, T : X → X. If there exist mappings ∝, β, γ, δ : X×X → [0,1) such that for all x, y ∈X :
(a) ∝(T Sx, y)≤∝(x, y)and ∝(x, ST y)≤∝(x, y),
β(T Sx, y)≤β(x, y)and β(x, ST y)≤β(x, y),
γ(T Sx, y)≤γ(x, y)and γ(x, ST y)≤γ(x, y),
δ(T Sx, y)≤δ(x, y)and δ(x, ST y)≤δ(x, y);
(b) ∝(x, y) +β(x, y) + 2γ(x, y) + 2sδ(x, y)<1;
(c)d(Sx, T y)-∝(x, y)d(x, y) + β(x,y)[1+1+d(dx,Sx(x,y))]d(y,T y)
+γ(x, y)[d(x, Sx) +d(y, T y)]+δ(x, y)[d(x, T y) +d(y, Sx)]. (2.2)
Then S and T have a unique common fixed point.
Proof. Let x, y ∈X, from (2.2) we have
d(Sx, T Sx)-∝(x, Sx)d(x, Sx) + β(x,Sx)[1+1+d(dx,Sx(x,Sx)])d(Sx,T Sx)
+γ(x, Sx)[d(x, Sx) +d(Sx, T Sx)]
+δ(x, Sx)[d(x, T Sx) +d(Sx, Sx)]
-∝(x, Sx)d(x, Sx) +β(x, Sx)d(Sx, T Sx)
+sδ(x, Sx)[d(x, Sx) +d(Sx, T Sx)]
which implies that
|d(Sx, T Sx)| ≤∝(x, Sx)|d(x, Sx)|+β(x, Sx)|d(Sx, T Sx)|
+γ(x, Sx)|d(x, Sx) +d(Sx, T Sx)|
+sδ(x, Sx)|d(x, Sx) +d(Sx, T Sx)|. (2.3)
Similary, from (2.2) we have,
d(ST y, T y)-∝(T y, y)d(T y, y) + β(T y,y)[1+1+d(dT y,ST y(T y,y))]d(y,T y)
+γ(T y, y)[d(T y, ST y) +d(y, T y)]
+δ(T y, y)[d(T y, T y) +d(y, ST y)]
which implies that
|d(ST y, T y)| ≤∝(T y, y)|d(T y, y)|+β(T y, y)|1 +d(T y, ST y)||1+d(dy,T y(y,T y))|
+γ(T y, y)|d(T y, ST y) +d(y, T y)|+sδ(T y, y)|d(y, T y) +d(T y, ST y)|
|d(ST y, T y)| ≤∝(T y, y)|d(T y, y)|+β(T y, y)|d(ST y, T y)|
+γ(T y, y)|d(ST y, T y) +d(T y, y)|
+sδ(T y, y)|d(ST y, T y) +d(T y, y)|. (2.4)
Letx0 ∈X and the sequence {xn} be defined by (2.1).
We show that {xn} is a Cauchy sequence. From Propostion 2.1 and inequalities (2.3), (2.4)
and for allK = 0,1,2,− − − − −, we obtain
≤∝(T x2K−1, x2K−1)|d(T x2K−1, x2K−1)|
+β(T x2K−1,x2K−1)|1+d(T x2K−1,ST x2K−1)||d(x2K−1,T x2K−1)|
|1+d(T x2K−1,x2K−1)|
+γ(T x2K−1, x2K−1)|d(T x2K−1, ST x2K−1) +d(x2K−1, T x2K−1)|
+δ(T x2K−1, x2K−1)|d(T x2K−1, T x2K−1) +d(x2K−1, ST x2K−1)|
=∝(x2K, x2K−1)|d(x2K, x2K−1)|
+β(x2K,x2K−1)|1+d(x2K,x2K+1)||d(x2K−1,x2K)|
|1+d(x2K,x2K−1)|
+γ(x2K, x2K−1)|d(x2K, x2K+1) +d(x2K−1, x2K)|
+δ(x2K, x2K−1)|d(x2K, x2K) +d(x2K−1, x2K+1)|
|d(x2K+1, x2K)| ≤∝(x2K, x2K−1)|d(x2K, x2K−1)|
+β(x2K, x2K−1)|d(x2K+1, x2K)|
+γ(x2K, x2K−1)|d(x2K+1, x2K) +d(x2K, x2K−1)|
+sδ(x2K, x2K−1)|d(x2K+1, x2K) +d(x2K, x2K−1)|
≤∝(x0, x2K−1)|d(x2K, x2K−1)|
+β(x0, x2K−1)|d(x2K+1, x2K)|
+γ(x0, x2K−1)|d(x2K+1, x2K) +d(x2K, x2K−1)|
+sδ(x0, x2K−1)|d(x2K+1, x2K) +d(x2K, x2K−1)|
+γ(x0, x1)|d(x2K+1, x2K) +d(x2K, x2K−1)|
+sδ(x0, x1)|d(x2K+1, x2K) +d(x2K, x2K−1)|
which yeilds that
|d(x2K+1, x2K)| ≤
∝(x0,x1)+γ(x0,x1)+sδ(x0,x1)
1−β(x0,x1)−γ(x0,x1)−sδ(x0,x1)|d(x2K, x2K−1)|.
Similarly, one can obtain
|d(x2K+2, x2K+1)| ≤
∝(x0,x1)+γ(x0,x1)+sδ(x0,x1)
1−β(x0,x1)−γ(x0,x1)−sδ(x0,x1)|d(x2K+1, x2K)|.
Letµ= ∝(x0,x1)+γ(x0,x1)+sδ(x0,x1)
1−β(x0,x1)−γ(x0,x1)−sδ(x0,x1) <1.
Since ∝(x0, x1) +β(x0, x1) + 2γ(x0, x1) + 2sδ(x0, x1)<1, thus we have
|d(xn+1.xn+2)| ≤µ|d(xn, xn+1)| ≤ − − − ≤µn+1|d(x0, x1)|. (2.5)
Thus for anym > n, m, n∈N,
|d(xn, xm)| ≤s|d(xn, xn+1)|+s|d(xn+1, xm)|
≤s|d(xn, xn+1)|+s2|d(xn+1, xn+2)|+s2|d(xn+2, xm)|
− − − − − − − − − − −
− − − − − − − − − − −
≤s|d(xn, xn+1)|+s2|d(xn+1, xn+2)|+s3|d(xn+2, xn+3)|
+− − − −+sm−n−1|d(x
m−2, xm−1)|+sm−n|d(xm−1, xm)|.
|d(xn, xm)| ≤sµn|d(x0, x1)|+s2µn+1|d(x0, x1)|+s3µn+2|d(x0, x1)|
+− − − −+sm−n−1µm−2|d(x0, x1)|+sm−1µm−1|d(x0, x1)|
=
m−n
X
i=1
siµi+n−1|d(x 0, x1)|.
Therefore,
|d(xn, xm)| ≤ m−n
X
i=1
si+n−1µi+n−1|d(x0, x1)|
=
m−1
X
t=n
stµt|d(x
0, x1)| ≤ ∞
X
t=n
(sµ)t|d(x 0, x1)|
=(1sµ−sµ)n|d(x0, x1)|
and so,
|d(xn, xm)| ≤ (sµ)
n
1−sµ|d(x0, x1)| →0 as m, n→ ∞.
This implies that {xn} is a Cauchy sequence in X. Since X is complete, there exists some
u∈ X such that xn→u as n → ∞. Let on contrary u6=Su, then there exists z ∈X such
that
|d(u, Su)|=|z|>0. (2.6)
So by using the triangular inequality and (2.2), we get
z =d(u, Su)
-sd(u, x2n+2) +sd(x2n+2, Su)
=sd(u, x2n+2) +sd(T x2n+1, Su)
-sd(u, x2n+2) +s ∝(u, x2n+1)d(u, x2n+1)
+sβ(u,x2n+1)[1+d(u,Su)]d(x2n+1,T x2n+1)
+sγ(u, x2n+1)[d(u, Su) +d(x2n+1, T x2n+1)]
+sδ(u, x2n+1)[d(u, T x2n+1) +d(x2n+1, Su)]
=sd(u, x2n+2) +s∝(u, x2n+1)d(u, x2n+1)
+sβ(u,x2n+1)[1+d(u,Su)]d(x2n+1,x2n+2)
1+d(u,x2n+1)
+sγ(u, x2n+1)[d(u, Su) +d(x2n+1, x2n+2)]
+sδ(u, x2n+1)[d(u, x2n+2) +d(x2n+1, Su)]
-sd(u, x2n+2) +s∝(u, x1)d(u, x2n+1)
+sβ(u,x1)[1+d(u,Su)]d(x2n+1,x2n+2)
1+d(u,x2n+1)
+sγ(u, x1)[d(u, Su) +d(x2n+1, x2n+2)]
+sδ(u, x1)[d(u, x2n+2) +d(x2n+1, Su)].
This implies that
|z| ≤ |d(u, Su)| ≤s|d(u, x2n+2)|+s∝(u, x1)|d(u, x2n+1)|
+sβ(u,x1)|[1+d(u,Su)]||d(x2n+1,x2n+2)|
|1+d(u,x2n+1)|
+sγ(u, x1)|z+d(x2n+1, x2n+2)|
+sδ(u, x1)|d(u, x2n+2) +z|.
|z| ≤s[γ(u, x1) +δ(u, x1)]|z|
≤s[∝(u, x1) +β(u, x1) + 2γ(u, x1) + 2δ(u, x1)]|z|
<|z|,
a contradiction and so |z|= 0, that is u=Su.It follows similarly that u=T u.
We now show that S and T have unique common fixed point. For this, assume that u?inX
is another common fixed point of S and T. Then
d(u, u?) =d(Su, T u?)
-∝(u, u?)d(u, u?) + β(u,u?)[1+d(u,Su)]d(u?,T u?) 1+d(u,u?)
+γ(u, u?)[d(u, Su) +d(u?, T u?)]
+δ(u, u?)[d(u, T u?) +d(u?, Su)]
-[∝(u, u?) + 2δ(u, u?)]d(u, u?).
Therefore, we have
|d(u, u?)| ≤[∝(u, u?) + 2δ(u, u?)]|d(u, u?)|.
Since ∝ (u, u?) + 2δ(u, u?) < 1, we have |d(u, u?)| = 0. Thus u = u?, which proves the
uniqueness of common fixed point in X. This completes the proof of the Theorem. By puttingS =T in Theorem 2.3, we deduce the following corollary.
Corollary 2.4. Let (X, d) be a complete complex valued b-metric space with the coefficient
s ≥1 and let T :X →X. If there exist mappings ∝, β, γ, δ :X×X → [0,1) such that for allx, y ∈X :
β(T x, y)≤β(x, y)and β(x, T y)≤β(x, y),
γ(T x, y)≤γ(x, y)and γ(x, T y)≤γ(x, y),
δ(T x, y)≤δ(x, y)and δ(x, T y)≤δ(x, y);
(b) ∝(x, y) +β(x, y) + 2γ(x, y) + 2sδ(x, y)<1;
(c) d(T x, T y)-∝(x, y)d(x, y) + β(x,y)[1+1+d(dx,T x(x,y))]d(y,T y)
+γ(x, y)[d(x, T x) +d(y, T y)]
+δ(x, y)[d(x, T y) +d(y, T x)]. (2.7)
Then T has a unique fixed point.
The following theorem is closely related to Corollary 2.4 with γ =δ= 0.
Theorem 2.5. Let (X, d) be a complete complex valued b-metric space with the coefficient
s ≥ 1 and let T : X → X. If there exist mappings ∝, β : X×X → [0,1) such that for all
x, y ∈X :
(a) ∝(T x, y)≤∝(x, y)and ∝(x, T y)≤∝(x, y),
β(T x, y)≤β(x, y)and β(x, T y)≤β(x, y);
(b) s{∝(x, y) +β(x, y)}<1;
(c) d(T x, T y)-∝(x, y)d(x, y) +β(x, y)d(y,T y1+)[1+d(x,yd(x,T x) )]. (2.8)
Proof. Let x0 ∈ X and the sequence {xn} be defined by xn+1 = T xn, where n =
0,1,2,− − − − − − −. (2.9)
We show that {xn} is a Cauchy sequence. From (2.8) we have
d(xn+1, xn+2) =d(T xn, T xn+1)
-∝(xn, xn+1)d(xn, xn+1)
+β(xn, xn+1)
d(xn+1,T xn+1)[1+d(xn,T xn)] 1+d(xn,xn+1)
=∝(xn, xn+1)d(xn, xn+1)
+β(xn, xn+1)
d(xn+1,xn+2)[1+d(xn,xn+1)]
1+d(xn,xn+1)
=∝(xn, xn+1)d(xn, xn+1) +β(xn, xn+1)d(xn+1, xn+2).
It follows from (a) that
d(xn+1, xn+2)-∝(xn, xn+1)d(xn, xn+1) +β(xn, xn+1)d(xn+1, xn+2)
-∝(x0, xn+1)d(xn, xn+1) +β(x0, xn+1)d(xn+1, xn+2)
-∝(x0, x0)d(xn, xn+1) +β(x0, x0)d(xn+1, xn+2).
Therefore
|d(xn+1, xn+2)| ≤∝(x0, x0)|d(xn, xn+1)|+β(x0, x0)|d(xn+1, xn+2)|
and hence
|d(xn+1, xn+2)| ≤
∝(x0,x0)
1−β(x0,x0)|d(xn, xn+1)|, for all n= 0,1,2,− − − −.
Since s{∝ (x0, x0) +β(x0, x0)} <1 and s ≥1, we get ∝(x0, x0) +β(x0, x0)<1. Therefore with µ= ∝(x0,x0)
1−β(x0,x0) <1,we have
n
completeness of X, there exists some u ∈ X such that xn → u as n → ∞. Next, we show
that u is a fixed point ofT. Then
d(u, T u)-d(u, T xn) +d(T xn, T u)
-d(u, xn+1)+∝(xn, u)d(xn, u)
+β(xn, u)d(u,T u1+)[1+d(xdn(x,un),T xn)]
-d(u, xn+1)+∝(x0, u)d(xn, u)
+β(x0, u)d(u,T u1+)[1+d(dx(xn,xn+1)] n,u) .
Letting n→ ∞, it follows that
|d(u, T u)| ≤β(x0, u)|d(u, T u)|,
a contradiction and so |d(u, T u)|= 0 that is u=T u.
Finally, we show the uniqueness. Suppose that there is u? ∈X such that u? =T u?.Then
d(u, u?) =d(T u, T u?)
-∝(u, u?)d(u, u?) +β(u, u?)d(u?,T u?)[1+d(u,T u)] 1+d(u,u?)
=∝(u, u?)d(u, u?)
and so d(u, u?) = 0, since ∝(u, u?) <1. This implies that u? = u, completing the proof of
the theorem.
3. Deduced results
Theorem 3.1. Let (X, d) be a complete complex valued b-metric space with the coefficient
s ≥ 1 and let S, T : X → X. If there exist mappings ∝, β : X → [0,1) such that for all
x, y ∈X :
(a) ∝(Sx)≤∝(x)and β(Sx)≤β(x),
∝(T x)≤∝(x)and β(T x)≤β(x);
(b) s{∝(x) +β(x)}<1;
(c) d(Sx, T y)-∝(x)d(x, y) + β(x)d1+(x,Sxd(x,y)d()y,T y). (3.1)
Then S and T have a unique common fixed point.
Proof. Define λ, µ:X×X →[0,1) by
λ(x, y) =∝(x) andµ(x, y) =β(x), for all x, y ∈X. (3.2)
Then for all x, y ∈X,
(a) λ(T Sx, y) =∝(T Sx)≤∝(Sx)≤∝(x) =λ(x, y) and
λ(x, ST y) =∝(x) = λ(x, y),
µ(T Sx, y) =β(T Sx)≤β(Sx)≤β(x) = µ(x, y) and
µ(x, ST y) =β(x) =µ(x, y);
(b) s{λ(x, y) +µ(x, y)}=s{∝(x) +β(x)}<1;
(c) d(Sx, T y)-∝(x)d(x, y) + β(x)d1+(x,Sxd(x,y)d()y,T y)
common fixed point.
The following Corollary is obtained from our Theorem 2.3.
Corollary 3.2. Let (X, d) be a complete complex valued b-metric space with the coefficient
s≥1 and let S, T :X →X. If there exist mappings ∝, β, γ, δ :X →[0,1) such that for all
x, y ∈X :
(a) ∝(T Sx)≤∝(x), β(T Sx)≤β(x), γ(T Sx)≤γ(x) and
δ(T Sx)≤δ(x);
(b) ∝(x) +β(x) + 2γ(x) + 2sδ(x)<1;
(c) d(Sx, T y)-∝(x)d(x, y) + β(x)[1+1+d(x,Sxd(x,y)])d(y,T y)
+γ(x)[d(x, Sx) +d(y, T y)]
+δ(x)[d(x, T y) +d(y, Sx)]. (3.3)
Then S and T have a unique common fixed point.
Proof. Define ∝, β, γ, δ :X×X →[0,1) by
∝(x, y) =∝(x), β(x, y) = β(x), γ(x, y) =γ(x) andδ(x, y) = δ(x),for allx, y ∈X. (3.4)
Then for all x, y ∈X,
(a) ∝(T Sx, y) =∝(T Sx)≤∝(x) =∝(x, y) and ∝(x, ST y) =∝(x) =∝(x, y),
β(T Sx, y) =β(T Sx)≤β(x) = β(x, y) and β(x, ST y) =β(x) =β(x, y),
δ(T Sx, y) = δ(T Sx)≤δ(x) =δ(x, y) andδ(x, ST y) = δ(x) =δ(x, y);
(b) ∝(x, y) +β(x, y) + 2γ(x, y) + 2sδ(x, y) =∝(x) +β(x) + 2γ(x) + 2sδ(x)<1;
(c) d(Sx, T y)-∝(x)d(x, y) + β(x)[1+1+d(x,Sxd(x,y)])d(y,T y)
+γ(x)[d(x, Sx) +d(y, T y)]
+δ(x)[d(x, T y) +d(y, Sx)]
=∝(x, y)d(x, y) + β(x,y)[1+1+d(dx,Sx(x,y))]d(y,T y)
+γ(x, y)[d(x, Sx) +d(y, T y)] +δ(x, y)[d(x, T y) +d(y, Sx)]. (3.5)
By Theorem 2.3, S and T have a unique common fixed point.
Letting ∝(.) =∝, β(.) = β, γ(.) = γ and δ(.) =δ in Corollary 3.2 gives the following result proved by Nashine and Fisher [9] (Complex valued b-metric space version).
Corollary 3.3. If S and T are self-mappings defined on a complex valued b-metric space (X, d) with the coefficient s≥1 satisfying the condition
d(Sx, T y)-∝d(x, y) + β[1+d1+(x,Sxd(x,y)]d)(y,T y)
+γ[d(x, Sx) +d(y, T y)]
+δ[d(x, T y) +d(y, Sx)] (3.6)
for all x, y ∈X, where ∝, β, γ and δ are nonnegative reals with∝+β+ 2γ+ 2sδ < 1. Then
S and T have a unique common fixed point.
Competing interests
The authors are thankful to the learned referees for their careful reading and critical remarks to improve the paper.
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